Past

Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients

possessing unique solutions make up a very important class in applications. For

instance, Langevin-type equations and gradient systems with noise belong to this

class. At the same time, most numerical methods for SDEs are derived under the

global Lipschitz condition. If this condition is violated, the behaviour of many

standard numerical methods in the whole space can lead to incorrect conclusions.

This situation is very alarming since we are forced to refuse many effective

methods and/or to resort to some comparatively complicated numerical procedures.

We propose a new concept which allows us to apply any numerical method of weak

approximation to a very broad class of SDEs with nonglobally Lipschitz

coefficients. Following this concept, we discard the approximate trajectories

which leave a sufficiently large sphere. We prove that accuracy of any method of

weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be

made arbitrarily small with increasing the radius of the sphere. The results

obtained are supported by numerical experiments. The concept of rejecting

exploding trajectories is applied to computing averages with respect to the

invariant law for Langevin-type equations. This approach to computing ergodic

limits does not require from numerical methods to be ergodic and even convergent

in the nonglobal Lipschitz case. The talk is based on joint papers with G.N.

Milstein.

A classical result due to Kunita says that if the coefficients are global

Lipschitzian, then the s.d.e defines a global flow of homeomorphisms. In this

talk, we shall prove that under suitable growth on Lipschitz constants, the sde

define still a global flow.

I aim to give some review of how generalised geometries provide a natural

framework for describing supersymmetric string backgrounds. In particular I

will focus on a rewriting of type II supergravity in terms of generalised

structures. Hitchin functions appear naturally along with generalised

extensions of the Gukov-Vafa-Witten superpotential.

In this talk, a network topology is presented that allows both peer-to-peer and non-local traffic in a cellular based TDD-CDMA system known as opportunity driven multiple access (ODMA). The key to offering appropriate performance of peer-to-peer communication in such a system relies on the use of a routing algorithm which minimises interference. This talk will discuss the constraints and limitations on the capacity of such a system using a variety of routing techniques. A congestion based routing algorithm will be presented that attempts to minimize the overall power of the system as well as providing a measure of feasibility. This technique provides the lowest required transmit power in all circumstances, and the highest capacity in nearly all cases studied. All of the routing algorithms considered allocate TDD time slots on a first come first served basis according to a set of pre-defined rules. This fact is utilised to enable the development of a combined routing and resource allocation algorithm for TDD-CDMA relaying. A novel method of time slot allocation according to relaying requirements is then developed.

Two measures of assessing congestion are presented based on matrix norms. One is suitable for a current interior point solution, the other is more elegant but is not currently suitable for efficient minimisation and thus practical implementation.

Image Inpainting turns the art of image restoration, retouching, and disocclusion into a computer-automated trade. Mathematically, it may be viewed as an interpolation problem in BV, SBV, or other fancy function spaces thought suitable for digital images. It has recently drawn the attention of the numerical PDE community, which has led to some impressive results. However, stability restrictions of the suggested explicit schemes so far yield computing times that are next to prohibitive in the realm of interactive digital image processing. We address this issue by constructing an appropriatecontinuous energy functional that combines the possibility of a fast discrete minimization with high perceptible quality of the resulting inpainted images.

The talk will survey the background of the inpainting problem and prominent PDE-based methods before entering the discussion of the suggested new energy functional. Many images will be shown along the way, in parts with online demonstrations.

This is joint work with my student Thomas März.

We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurrence of nonuniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we obtain a first ever explicit lower bound for the set of parameters corresponding to maps in the quadratic family f_{a}(x) = x^{2}-a which have an absolutely continuous invariant probability measure.